Was not expecting that at the start 😂

@RS-go2sn

2 жыл бұрын

台湾！

This guy also speaks Chinese, Grant has everything covered.

@ilickcatnip

2 жыл бұрын

I'm absolutely blown away by this man!

@gasun1274

2 жыл бұрын

his tones are off though, but i never spoke mandarin in a colloquial setting so maybe there's something about when it's ok to change tones in certain contexts.

@alsatusmd1A13

2 жыл бұрын

@@donsanderson Mandarin is what people commonly mean by the language “Chinese”.

@seeker4430

2 жыл бұрын

@@donsanderson Really where did you find this information? I really want to hear him speak chinese

@user-zu1ix3yq2w

2 жыл бұрын

This is the advantage of a powerful memory.

To any of you wondering, the outro (下次一定) is a common Chinese phrase/meme to say "maybe next time" or "take a rain check".

@mrnarason

2 жыл бұрын

Basically never lol

@dqrksun

2 жыл бұрын

下次亿定

@blue-whitestar2598

2 жыл бұрын

Yes, great pun, wasn’t expecting that!

@esepecesito

2 жыл бұрын

Like the mexican "Ahorita!" ?

@superlugiaultra

2 жыл бұрын

'next time for sure'

i will now write all my texts and articles with the following notation: :) signifies the end of a proof :/ signifies a question :( signifies a limitation in the experiment edit: :o signifies a theorem or lemma

@yurigouveawagner9432

2 жыл бұрын

god dang it I edited and lost the heart

@nealcarpenter3093

2 жыл бұрын

I really like your notation - will be suggesting it to my students for their notes. Thanks.

@thedoublehelix5661

2 жыл бұрын

I like to draw a smiley on the inside of the end of proof square

@yurigouveawagner9432

2 жыл бұрын

@@nealcarpenter3093 thank you! are you a professor? It's cool to imagine this idea influencing a class somewhere in the world

@nealcarpenter3093

2 жыл бұрын

@@yurigouveawagner9432 Hey, good or humorous (if there's a diff.) ideas spread. I teach math, grades 6-8, though that does include some Calculus.

12:00 this is why Grant is among the best maths educators out there. He talks about the emotion in maths, about stressing what's important vs the detail. It explains why the first time I properly intuited linear independence was after looking at his Linear algebra series. Some people will argue you can't visualise all maths but I think there's a happy medium between slavishly following symbols on a page and characterising what they represent more fully. This is why Grant is a bit of a hero for me.

@MathTutor1

2 жыл бұрын

Agree.

@Notsosarcastic_02

2 ай бұрын

Well just to tell as a fact , here in India , textbooks do have these kind of emojis , more like characters as whole in textbooks interacting with children and trying to tell them the concepts like a story or something through the textbook . I remember here in my state board textbook (local educational board's published textbook ) , there was a Rubik's cube interacting with us asking us questions or giving us a recap of the previous chapter at the start of a new chapter and also giving us some tips , they put him into corners and it was interesting.

I would recommend Jiří Matoušek's "Thirty-three miniatures". It's about 33 beautiful applications of linear algebra. Also, all textbooks by him that I checked out are gold.

@GrantSanderson

2 жыл бұрын

Sounds amazing, I’ll check it out!

@VarunRamanathan28031999

2 жыл бұрын

+inf Matousek's books are incredible!

Great video! Really hope you keep posting on this channel. (The podcast is also fantastic!)

Dude, you're a legend. Thank you for the hard work you put into your videos to make things clearer. Seriously, thank you!!

You are an extremely talented communicator, for real.

Being a practicing engineer, I'm really interested in theory of nonlinear dynamics and mathematics behind. The Nonlinear Dynamics Book will be a real help. Thanks very much, Grant! ❤

Please do not stop making those videos 🙏🙏 You are changing my life 🙂

Fastest 25 mins of my life. Love listening to you Grant, always a pleasure.

Hubbard’s book is fantastic! It was the textbook for my first proof-based class, and it made me switch to the math major

Great video. Agreed on visualization as a gateway to help. I have some stuff on symmetry and computation. Have a great day and keep up the great work.

When talking about natural talent, I think about crafting. When I have just started teaching someone to knit or crochet, at some point, after they've gotten just good enough to reliably make stitches, they'll watch me do the same craft and say something like "gosh you're so fast and I'm so slow, I wish I was fast". And when it's about crafting, the response to this is actually really intuitive. I say "well, I've been doing this about two decades longer than you have, so I should be faster; if I wasn't, there would be something going rather wrong".

@kreek22

2 жыл бұрын

But, bio-communism isn't real.

Got some books for Christmas this year tbh -- pretty excited thanks for the recommendations

Love your work Grant ❤❤❤

that question about not being gifted enough was hard to hear. coming from a competition math background, i often felt that the community could be very toxic and make me feel like i was not "gifted" enough

@HAL--vf6cg

2 жыл бұрын

@@markv785 this is just my opinion, but SATs are a terrible mode of evaluation, especially for college admissions.

Ahhhhh! Didn’t expect this at all lol. Love your contents!

Competition math can be really discouraging for students who want to get into math but can’t “keep up” with everyone else. I never even thought that I could ponder problems for a day or more. Everything was kinda fast-paced. It’s funny that continuity can work perfectly fine only using rational numbers. Talk about lifting up your pencil! Oh and that room problem, I didn’t find that easy at all lol. Although, convex functions ARE quite interesting *cough* *cough* Legendre transform. Love your content! Can’t wait for more on diff eqs and probability

@xXDarQXx

2 жыл бұрын

the room problem really isn't that hard. click read more if you want a hint (to find the solution I came up with) think about the room with the smallest number of people.

@kreek22

2 жыл бұрын

The agon is all.

Hey! I thought of the sum of squares solution almost instantly. That feels pretty neat

I have been following and looking up to this guy for years, and never would have expected him to speak Chinese so fluently.

Thank you, sir! I come from China, and I learned quite a lot from your teachings. They are very insightful! Thank you very much! 感谢！

@Benderhino

2 жыл бұрын

is china fun?

@user-zx6os1fy6w

2 жыл бұрын

@@Benderhino Why don't you come to China and get the answer by yourself?

@Benderhino

2 жыл бұрын

@@user-zx6os1fy6w nah man busy with poverty stuff, this is embarrassing I wish everyone was dead, anyways, have a nice day.

I've always enjoyed math and physics. The one thing I have learned over time is that numbers, math and all that follows they are nothing more than a product of the mind as they are all conceptual ideas. At the end of the day, math is nothing more than a model and it is what you make of it! I enjoy hearing all different kinds of perspectives and approaches. 3Blue1Brown is one of my favorites along with Mathologer and a few others... I love their presentations and visualization methods. It's almost like comparing or listening to a monotone college professor who'd soon enough put you to sleep on trying to explain the phenomenons of the Mandelbrot Set as opposed to writing a simple computer program to graph it visually being able to see and interact with the infinite patterns and levels of details that a simple equation produces. Grant and others like him allow mathematics to be fun and engaging instead of being monotonous, drab and dreary... Keep up the great work!

For the room problem: a "physical/intuitive" interpretation of the sum-of-squares is as a weighted count of the population, where each person is weighted by the number of their roommates.

@__koaaa_9619

2 жыл бұрын

Can’t help but mention that it can also be viewed as the variance of a distribution + the mean squared, where in this case the mean is constant, so it basically reflects the variance.

@DS-qg9cd

2 жыл бұрын

I like to think that it is because of the fact that (x + a)^n > x^n + a^n, you can always choose x^n with n > 1 and you'll get the same proof

I love your Chanel 💕💕 You are the best!!

The Hubbard text is rather good for all the topics covered. I think the book is especially great for those who have studied already Linear Algebra, Real Analysis, and Multivariable Calculus and are in need of a more in depth review. We used that book for Math 23a & b with Paul Bamberg at Harvard. When I took the class, we used that very same 4th edition.

I am really enjoying these videos on this second channel. Great to hear conversations and interviews that give more depth behind the characters in mathematics. On the nature vs nurture aspect. I really do feel that there is an in-built ability to engage with mathematics in some people. I know that I saw friends with whom I performed on level with in undergrad really outpace me in graduate courses. It really showed me there was a difference in ability. There is a spectrum of human ability and Tao and Von Neumann are really at the peak of this. If there are people at the top and the bottom, there are going to be people at every point in between. Of course, this is really hard to quantify, and I don’t think there is value in approaching education this way. I think most topics taught in school should be approachable to everyone, but after a point, you see real differentiation between students.

@GrantSanderson

2 жыл бұрын

What’s hard to tell is if this is actually innate, or based on instincts learned so young that it might as well be innate.

@JoelRosenfeld

2 жыл бұрын

@@GrantSanderson yeah, it really is hard to characterize and identify. Though it is undeniable with people like Gauss, where they comprehended arithmetic at 3 years old. But that may be apocryphal too.

That intro woke me up!

Thank you soooo much, I love your videos!

我从很早的时候就开始看你的影片，这么长时间一直都很喜欢你的影片。你给了我很多对于数学新的想法。我也很喜欢manim。

2:09 actually during my third year in university I had a course which was dedicated only to proofs from this book. Thank you prof. Yuval roichman!

I bet whoever came up with that sum of squares solution had some connection to Cambridge. They used to grade their exams by taking the sum of squares of the scores to each question, to encourage people to prioritise answering fewer questions more completely rather than doing a lot of questions partially. As you say, problem solving is a lot about your experience....

Wow, what a beautiful narrative voice! So glad you didn’t dumb this one down by putting music over this fascinating Information 😇👍

Thank you for this video.

22:13 Being a majorly doubtful visual learner of mathematics, I've been dying to know this. Thank you so much Grant .

I would say that as long as you've done a reasonable amount of preparation then the fault pretty much always lies with the textbook. One of many examples of this is I once supported ten different students through the exact same maths course. 7 had textbook A - Topic X was impenetrable, only ever to be understood by the best and the brightest. Topic Y was completely straightforward and I was never asked about it. 3 had textbook B - Topic X was completely straightforward and I was never asked about it. Topic Y was impenetrable, only ever to be understood by the best and the brightest. I taught Topic X to those with textbook A, and Topic Y to those with textbook B, made those topics seem completely straightforward and everyone thought I was a genius.

I was moving stuff around and now my computer screen sits closer to me. Finally understand why your channel is called 3 Blue 1 Brown! :D

@Gameboygenius

2 жыл бұрын

It's always amazing the first time you notice!

Heard the lil intro theme phrase and couldn't help but to think on 3Blew1Blown

waiting for your next videos, just the old good 3b1b

6:47 Regarding the concept of being gifted vs. not: I've always held to the belief that there *are* people who are better in a given field by their nature, but *anyone* can become skilled through effort. Anyone can become skilled, but some people have a head start. Regarding math specifically, yes. There are individuals who are naturally good at it; their brains are wired in a way that makes learning mathematics natural. These are the individuals we'd typically call "gifted." For others, learning math is harder. The ""not gifted"" individuals. It's very easy to see someone breeze through things that you struggle with and come to the conclusion that you're just not gifted enough to progress, but that's categorically untrue. Just because it takes more effort for you to learn something doesn't mean you can't. If you're willing to put your head down and push through, you can learn any skill you want. Whether you're gifted or not determines where you start, not where you end up.

@geraldsnodd

2 жыл бұрын

Agreed, I think the same.

@NoriMori1992

2 жыл бұрын

I'm not sure it's true that _anyone_ can become skilled at math just by putting in more effort. Frankly, I don't think that's true of _anything._ But I agree that there are most likely many people in the world who could become skilled at math, but are never given a chance or a reason to put in enough work.

@Adhjie

8 ай бұрын

Given a chance: ramanujan. Or has the chance Richard Feynman said hard works he didn't want honors.

@Adhjie

8 ай бұрын

Tho yes genes is a factor. Eg cofactor factor constanta etc iirc l'hospital rule

5:00 Yes yes yes yes, I am obsessed with Valadmir Arnold's writings, I have been collecting his works in English but most of his writings are in Russian. Currently learning a lot from his book on Catastrophe Theory. I feel VI Arnold changed my life, his essays on mathematical pedagogy are revolutionary and I feel you are the one who will make his dream come true of unifying maths of physics which were divorced by the Evil Bourbaki's.

@maalikserebryakov

2 жыл бұрын

Lol don’t let arnold brainwash you with his nonsensical opinions the bourbaki’s did a great job and the existence of Pure mathematics is super convenient and useful

So, you created Manim!... you are the best!

This dude is amazing

2:25 woah! Grant, in your interview with Steven Strogatz, he mentioned teaching at a math program at Hampshire College (HCSSiM). I just realized today that turns out this quote from Erdos was actually in 1985 when he visited HCSSiM and was giving a talk. "yellow pig camp" :)

Recently read Mathematical Methods of Classical Mechanics by Vladimir Arnold and it's probably my favourite book. Super cool topics and ingeneous presentation, just a beautiful textbook.

is it weird that i almost cried at ( 6:47 - Not being "gifted" enough ) ?

@particleonazock2246

2 жыл бұрын

Why you no doctor??

Nonlinear dynamics was phenomenal

As for the continuity example. I think the most discrete answer would be a function is continuous at a given input when both of its limits exist and are equal or converge to the same value. Take sin(t) for example, no matter what input you use within this function both the left and right hand limits exist and are equal. Now, take the tangent function and examine its left and right hand limits at tan(PI/2) or tan(45) and other multiples of them. Both limits exist, yet they are not equal. The limit approach towards the right tends to +infinity and the limit approach towards the left tends to -infinity. Here we can see that the sine and cosine functions are both continuous where the tangent, cotangent, secant and cosecant are not continuous for all of their domains. The challenging thing here is that we are typically taught about continuity with respect to functions and graphs before we are ever introduced to the concepts of limits. Yet I think the best proof and representation of continuity lies within the properties of a functions limits across all of its domain.

The question with the room made me think about the Herfindahl-Hirschmann that measures market concentration.

Dude, your eye IS the 3b1b logo! Never noticed it before! Mind = blown

This is full of valuable suggestions. Just "the convexity of x squared" at 21:20 led me to an answer to a question that's bothered me for years, namely: "Why do we nearly always use & teach Standard Deviation, rather than MAD or AAD?" Not bad for a 5 second toss off remark!

@columbus8myhw

2 жыл бұрын

It seems to me that you can add up other functions and it will still work, for example adding 2^x where x is the number of people for each room

@caiollvllal

2 жыл бұрын

@@columbus8myhw i think we just need to use a strictly convex function

A different way to think about the room problem: for each person, count how many people they share a room with (including themselves) and add all of those numbers up. It's easy to see that the result is the sum of squares of the people in each room. But the number increases for the person moving rooms, it increases by 1 in the people of the room they're moving into and decreases by 1 for the people in the room they're moving out of. By the fact that there are more people in the former, the result is a net increase in the total.

This video came up in my feed at EXACTLY the right moment. Just a couple of hours ago, I finally decided that the vexation of a pop-math book I was trying to learn from was too much for me to deal with, and it's time for me to move on, so as not stymie my progress. (I've always struggled with math since my algebra days, but aced logic, and have managed to teach myself enough of the relevant math for my physics to not fall too far behind.) I'm not going to say the book was 'bad'. Just that it's way of presenting the information--some stuff about a family of 6 boys, 6 girls, a mom and dad, and trying to see how many dance pairs the siblings can make--was not helpful at all. But at least I learned what FACTORIAL is!

19:30 The problem of rooms is very closely related to the Raising Operator on integer partitions

Thanks Grant. 👍

It's amazing that the first couple minutes of this are the reason I now know where the name "3Blue1Brown" comes from. Genuinely so damn cool.

Incredible

Grant giving a student friendly perspective on Mathematics, is probably on of the best things on KZread. What I am missing in the book recommendations list is a discrete mathematics book, is there any good textbook about the subject for autodidacts?

The problem at 19:00 really looks like some sort of inverse of entropy since entropy is a time dependent phenomina I think this is a very interesting problem.

Hello Grant. I am immensely benefiting from your videos. Can you please make a video where you could maybe show a roadman for self learners of mathematics? Your favorite books?

your right eye really is 3blue1brown

@writethatdown100

Жыл бұрын

wait what? that's crazy!

the intro got me shook

Hi Grant, was wondering if there are journals or sources of papers you regularly read? Also any recommendations on book cases?

18:31 Grant tries to remember a mathematical puzzle, his platonic brainworks appears to screen.

You mentioned that it would be great if there were emojis in the margins of textbooks to help with understanding. In the Park City Math Institute (PCMI) problem sets used in their math teacher seminars, there are those sorts of comments (sadly, no emojis) in the margins. Sometimes they’re there for humor, but other comments are informative. I really like them!

Very interesting.

Wishing to have a Calculus book from james stewart in my bday today

Being frustrated by my math textbooks is how I found this channel in the first place. 😛

Dear Grant, The one-third of Linear Algebra of the Hubbard and Hubbard book is a great resource of LA but is rather short. And the longer, more elaborate textbooks of Linear Algebra that I could find were unintuitive(Like Axler, Strang, etc). Books like Hoffman Kunze and Halmos seem to be at a higher level than what I currently am at. Are there other detailed textbooks of Linear Algebra that teach the subject primarily as the study of *linear transformations* rather than the study of mathematical objects consisting of a bunch of numbers arranged in an array and weird operations on those objects like multiplication whose rules kind of seem arbitrary, pointless and unmotivated.

@simply42

2 жыл бұрын

I just took a college course called "abstract linear algebra" and we used the first 4 chapters of a book called "linear algebra done wrong." Rather than just starting with the algorthims on vectors and matricies, the book starts from the most abstract definition of a vector space, then looks at transformations between vector spaces, then shows exactly why matricies characterize linear transformations. The book then uses that as a jumping off point to learn the algorthims and stuff like row reduction, rank, and eigenvalues. Based on your description, it sounds like it's what you are looking for

@martinepstein9826

2 жыл бұрын

I'd recommend Axler but it sounds like you already tried him. He's all about emphasizing linear transformations over matrices. What did you find unintuitive about his book?

@DD-rl7xo

2 жыл бұрын

@@simply42 Oh that sounds good, I would surely like to try that.

@DD-rl7xo

2 жыл бұрын

@@martinepstein9826 I like that Axler's book tells us abstract definitions rather than focusing on computations. But my problem with the book is that it feels less motivated. Throughout the book, all sorts of operations and mathematical objects with their different properties are introduced. However, we are never told once why mathematicians invented that concept (I am not so concerned with real life applications but I really want to know why mathematicians defined things in one way rather than the other. Surely, all of the definitions in the book were later formalizations of intuitive ideas of mathematicians. Those ideas that motivated them to define those objects with those properties in *that specific way and not any other* is what I want to know so that I can appreciate what the definition actually stands for). And Axler's book while being abstract, is not intuitive or motivated (atleast to me).

@nicholasandrzejkiewicz

2 жыл бұрын

Axler's book is not abstract lol

What's crazy is that i recently picked up each of the books Grant mentions except for the Cauchy schwarz master class

I don't believe in the philosophy that frustration is apart of your interest. The reason is that I have found plenty of scenarios where reading a book has been less or not at all frustrating compared to jumping blindly into a project. I enjoyed the not at all frustrating approach and I learned more from that.

the intro!, does grant know Chinese or did he just learn the greeting?

@GrantSanderson

2 жыл бұрын

I took some Chinese back in high school, and spent a couple weeks in Beijing. Sadly I've forgotten most. This was a greeting a translator handed to me, likewise for the outro.

@le0nz

2 жыл бұрын

@@GrantSanderson you are so gangster that I just assume you knew mandarin.

@microhoarray

2 жыл бұрын

@@le0nz ahahhahahahaha samee

Imagine Next time grant come up with "Namaste, To kese hen aap log" I hope someone from India will sponsor you for making some series.

@shrey377

2 жыл бұрын

@@quirtt HM orz 🛐

@srikrishna250

2 жыл бұрын

😂

You got me scared af at the beginning..lol

Thank you for uploading it here as well. I have a Bilibili account, but as an English speaker, KZread is a lot easier to use. Also uh, how does one find a good translator for mathematical content? Considering giving my SoME1 videos a Chinese translation down the road

Very nice

You can easily add one plus one is two. But real mechanical trace for that is a different story. There are so many ways but there is no proof that there is only one way to solve a problem. Though the idea is the same result. Because source and destination are same for every problem. Visualisation is the different forms of how human brains work. Almost 90% of mathematical solutions lie on filling the gaps. For example a random numbers sequence to an orderly is filling the gaps to regular.

probability series when? 下次一定! 下次一定!

I need a version of this video but every time he says "The Book" the volume goes way up

Hi Grant, what's your thoughts on Nathan Carter’s book "Visual Group Theory"? I like all the visualizations, applications and even warnings against misconceptions. Is there any better book recommendation for Group Theory?

@shawon265

2 жыл бұрын

I should probably mention I am currently learning Group Theory from this book. I have gone to an introductory class, but everything was just too abstract. I understood them, like Cosets for example. But those definitions didn't stick with me. But here, they use Cayley diagrams and point out Cosets with a few examples which creates a clear mental image for me. I worry if this book is not "Complete and concrete enough". But personally I am open to reading other books after this.

I've just realized the reason why the channel is called 3b1b - Grant's right eye has a mix of brown and blue. It's amazing how this detail only struck me today!

您的中文非常好!Your Chinese is very good!

12:12 The book, _Cryptological Mathematics_ by Robert Edward Lewand, has a few characters in it that react, and "interact" with the information.

I'm pleased to have come up with the same initial guess for how to approach the problem you described at 18:38 Normally I think that off the top of my head, if I'm trying to prove that something will eventually be the case, I can try to come up with some monovariant. Would I have come up with this in high school though? Probably not.

So cool

My recommendation: "Elements" from Euklid. First learn the basics and learn to think mathematically, then the rest is easy. It's like jumping in a pool. First, the water is just cold. After a while, it makes really fun.

@raptorzz8571

Жыл бұрын

Do you mean Euclid?

Can you make a video showing how you make your lecture videos for 3b1b? 🥺🥺🥺

I didn't really noticed it before but his right eye-ball is almost identical to the 3b1b logo, I know the some of the inspiration for the logo came from his eye but I didn't expect this much resemblance

I find it interesting that your channel reached 1M subscribers just after Steve Mould's and ahead of Matt Parker's (which hasn't happened yet, he's still around 900K). But all in all it's a great win for STEM.

@yonatanbeer3475

2 жыл бұрын

3blue1brown hit 1 million subs in 2018, I'm pretty sure that's well before Steve Mould

@1ogic948

2 жыл бұрын

His main KZread is at 3.85M right now, this video celebrates the Chinese channel (on BiliBili) reaching 1M! The rest of the translated channels have between a few hundred and a tens of thousands subs each.

What did he say in last in reply of probability series when? I'm not understanding.

for anyone interested, the lectures in Discrete Differential Geometry given by Keenan Crane on youtube are amazing.

Calculus is brain of Maths and Alzebra its heart

For the rooms puzzle, you could also add up 2^(x_i), right?

I only just now noticed how one of your eyes is 3/4 blue and 1/4 brown. Guess that's where the name comes from xD

@Derpuwolf

2 жыл бұрын

And the logo design too, it’s an iris!

I think I have a problem. I dig all the axioms and definitions when I start reading about a branch of mathematics but I don't really like the how all this is usefull part that comes after.

Is the Vector Calculate book a collector's item? Its list price on Amazon is $879.98.

when he spoke chinese in the intro i SCREAMED he did so well i love my bilingual king

@berryzhang7263

2 жыл бұрын

@@donsanderson yeah!! his pronunciation was so good and he even added some tones which is super impressive

HI grant, do you have elementary linear algebra book recommendation? I mean i need a book that give some important proof but not too early, they give the definition/lemma/theorem first, than they give some problem with number and answer, the proof of the theorem can come later, (other than anton howards book).

Grant Do a podcast with Physicist V.Balakrishnan